Using variational methods, we establish the existence and multiplicity of positive solutions for the following class of problems:

where λ,β∈(0,∞), q∈(1,2*−1), 2*=2N/(N−2), N≥3, V,Z:ℝN→ℝ are continuous functions verifying some hypotheses.

Given two unital continuous C*-bundles, A and B, over the same compact Hausdorff base space X, we study the continuity properties of their different amalgamated free products over C(X).

A problem posed in the early eighteenth century asks for right-angled triangles, each of whose sides exceeds double the area by a perfect square. We summarize known results and find such triangles with the smallest possible standard generators.

In this paper we determine the group of endotrivial modules for certain symmetric and alternating groups in characteristic p. If p = 2, then the group is generated by the class of Ωn(k) except in a few low degrees. If p > 2, then the group is only determined for degrees less than p2. In these cases we show that there are several Young modules which are endotrivial.

We study the sharp threshold for blow-up and global existence and the instability of standing wave eiωtuω(x) for the Davey–Stewartson system

in ℝ3, where uω is a ground state. By constructing a type of cross-constrained variational problem and establishing so-called cross-invariant manifolds of the evolution flow, we derive a sharp criterion for global existence and blow-up of the solutions to (DS), which can be used to show that there exist blow-up solutions of (DS) arbitrarily close to the standing waves.

We study a class of second-order nonlinear differential equations on a finite interval with periodic boundary conditions. The nonlinearity in the equations can take negative values and may be unbounded from below. Criteria are established for the existence of non-trivial solutions, positive solutions and negative solutions of the problems under consideration. Applications of our results to related eigenvalue problems are also discussed. Examples are included to illustrate some of the results. Our analysis relies mainly on topological degree theory.

Let Ω ⊂ ℝN be a bounded domain such that 0 ∈ Ω, N ≥ 3, 2*(s) = 2(N − s)/(N − 2), 0 ≤ s < 2, . We obtain the existence of infinitely many solutions for the singular critical problem with Dirichlet boundary condition for suitable positive number λ.

There are only finitely many non-constant holomorphic mappings between two fixed compact Riemann surfaces of genus greater than 1. This result goes under the name of the de Franchis theorem. Having seen that the set of such holomorphic mappings is finite, we naturally want to obtain a bound on its cardinality. It has been known for some time that there exist various bounds depending only on the genera of the surfaces. Here we obtain ‘better’ bounds of the above type, using arguments based on the rigidity of holomorphic mappings and the hyperbolic geometry of surfaces.

We analyse Hecke pairs (G,H) and the associated Hecke algebra when G is a semi-direct product N ⋊ Q and H = M ⋊ R for subgroups M ⊂ N and R ⊂ Q with M normal in N. Our main result shows that, when (G,H) coincides with its Schlichting completion and R is normal in Q, the closure of in C*(G) is Morita–Rieffel equivalent to a crossed product I⋊βQ/R, where I is a certain ideal in the fixed-point algebra C*(N)R. Several concrete examples are given illustrating and applying our techniques, including some involving subgroups of GL(2,K) acting on K2, where K = ℚ or K = ℤ[p−1]. In particular we look at the ax + b group of a quadratic extension of K.

We study commutation properties of subsets of right-angled Artin groups and trace monoids. We show that if Γ is any graph not containing a four-cycle without chords, then the group G(Γ) does not contain four elements whose commutation graph is a four-cycle; a consequence is that G(Γ) does not have a subgroup isomorphic to a direct product of non-abelian groups. We also obtain corresponding and more general results in the monoid case.

We consider semilinear elliptic problems in which the right-hand-side nonlinearity depends on a parameter λ > 0. Two multiplicity results are presented, guaranteeing the existence of at least three non-trivial solutions for this kind of problem, when the parameter λ belongs to an interval (0,λ*). Our approach is based on variational techniques, truncation methods and critical groups. The first result incorporates as a special case problems with concave–convex nonlinearities, while the second one involves concave nonlinearities perturbed by an asymptotically linear nonlinearity at infinity.

We complete the investigation of growth properties of analytic functions connected with the Nevanlinna parametrization of the solutions of an indeterminate strong Hamburger moment problem.

The theorem referred to in the title is a technical result that is needed for the classification of elliptic and K3 fibrations birational to Fano 3-fold hypersurfaces in weighted projective space. We present a complete proof of the curve exclusion theorem, which first appeared in the author's PhD thesis and has since been relied upon in solutions to many cases of the fibration classification problem. We give examples of these solutions and discuss them briefly.

Suppose that G is an abelian group and that A ⊂ G is finite and contains no non-trivial three-term arithmetic progressions. We show that |A+A| »ε|A|(log|A|)⅓−ε.

We prove that if two transcendental meromorphic functions share all limit values from a set of positive linear measure on a rectifiable Jordan arc, then they share all limit values.

Theorems on the Fredholm alternative and well-posedness of the characteristic initial-value problem

are established, where l : C(;ℝ) is a linear bounded operator, q ∈ L(;ℝ), ϕ:[a,b]→ℝ, ψ:[c,d]→ℝ are absolutely continuous functions and =[a,b]×[c,d]. Some solvability conditions of the problem considered are also given.

We use the compensated compactness method coupled with some basic ideas of kinetic formulation developed by Lions, Perthame, Souganidis and Tadmor to give a refined proof for the existence of global bounded entropy solutions to the Le Roux system. This new method of the reduction of Young measures can be applied to solve other problems.